Found problems: 2008
2007 Moldova Team Selection Test, 2
Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.
2000 Junior Balkan Team Selection Tests - Romania, 2
Let be a natural power of two. Find the number of numbers equivalent with $ 1 $ modulo $ 3 $ that divide it.
[i]Dan Brânzei[/i]
2001 JBMO ShortLists, 2
Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that:
i) In the set $P_n$ there are infinitely many squares.
ii) In the set $P_n$ there are no squares.
2013 Macedonia National Olympiad, 1
Let $ p,q,r $ be prime numbers. Solve the equation $ p^{2q}+q^{2p}=r $
2019 Danube Mathematical Competition, 1
Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.
2012 China Team Selection Test, 3
Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have
\[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \]
Find the number of [i]good[/i] functions.
1986 Spain Mathematical Olympiad, 3
Find all natural numbers $n$ such that $5^n+3$ is a power of $2$
2004 National Olympiad First Round, 26
What is the last two digits of base-$3$ representation of $2005^{2003^{2004}+3}$?
$
\textbf{(A)}\ 21
\qquad\textbf{(B)}\ 01
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 02
\qquad\textbf{(E)}\ 22
$
2009 Purple Comet Problems, 15
What is the remainder when $7^{8^9}$ is divided by $1000?$
1992 AMC 8, 15
What is the $1992^\text{nd}$ letter in this sequence?
\[\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\cdots \]
$\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$
2012 AMC 10, 20
Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$
2000 CentroAmerican, 2
Determine all positive integers $ n$ such that it is possible to tile a $ 15 \times n$ board with pieces shaped like this:
[asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]
PEN D Problems, 23
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list]
1982 AMC 12/AHSME, 28
A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased?
$\textbf{(A) } 6\qquad \textbf{(B) }7 \qquad \textbf{(C) }8 \qquad \textbf{(D) } 9\qquad \textbf{(E) }\text{cannot be determined}$
PEN D Problems, 9
Show that there exists a composite number $n$ such that $a^n \equiv a \; \pmod{n}$ for all $a \in \mathbb{Z}$.
2004 APMO, 4
For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that
\[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \]
is even for every positive integer $n$.
2013 Moldova Team Selection Test, 1
Let $A=20132013...2013$ be formed by joining $2013$, $165$ times. Prove that $2013^2 \mid A$.
2010 Hong kong National Olympiad, 3
Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that
\[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\]
where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.
1971 Bundeswettbewerb Mathematik, 2
You are given a piece of paper. You can cut the paper into $8$ or $12$ pieces. Then you can do so for any of the new pieces or let them uncut and so on.
Can you get exactly $60$ pieces¿ Show that you can get every number of pieces greater than $60$.
2006 Hong kong National Olympiad, 2
For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1}$ = $f_1 \circ f_n$ . Evaluate $f_{2007}(2^{2006} )$.
2010 Contests, 1
Solve the equation
\[ x^3+2y^3-4x-5y+z^2=2012, \]
in the set of integers.
2005 AMC 10, 15
How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 6$
2003 AIME Problems, 8
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.
2014 AMC 10, 17
What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$?
$ \textbf{(A) }2^{1002}\qquad\textbf{(B) }2^{1003}\qquad\textbf{(C) }2^{1004}\qquad\textbf{(D) }2^{1005} \qquad\textbf{(E) }2^{1006} \qquad $
2012 JBMO TST - Turkey, 2
Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.